Automorphism groups of p-groups

DefinitionBerkovich's problem on automorphism groups of p-groups (Kourovka 15.27)#✓fully proved

The statement that for some prime \(p\) there are a finite \(p\)-group \(G\) of order greater than \(2\) and a proper subgroup \(H < G\) whose automorphism groups are isomorphic, \(\mathrm{Aut}(G) \cong \mathrm{Aut}(H)\).

def autIsoAutOfProperSubgroup : Prop :=
  ∃ p : ℕ, p.Prime ∧ ∃ (G : Type) (_ : Group G) (_ : Finite G), IsPGroup p G ∧ 2 < Nat.card G ∧
    ∃ H : Subgroup G, H ≠ ⊤ ∧ Nonempty (MulAut G ≃* MulAut H)

Used by 1

Used byA p-group and a proper subgroup with isomorphic automorphism groups (Kourovka 15.27)

TheoremA p-group and a proper subgroup with isomorphic automorphism groups (Kourovka 15.27)#●ready to prove

For some prime \(p\) there are a finite \(p\)-group \(G\) of order greater than \(2\) and a proper subgroup \(H < G\) with \(\mathrm{Aut}(G) \cong \mathrm{Aut}(H)\) as groups.

T. Li's example is a \(2\)-group \(G\) of order \(32\) with a subgroup \(H\) of order \(16\).

public theorem_wanted autIsoAutOfProperSubgroup_holds : autIsoAutOfProperSubgroup

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 15.27 (Ya. G. Berkovich); resolution: T. Li, Arch. Math. 92 (2009), 287-290, Kourovka Archive, Problem 15.27 (answer: Yes, it is possible, for |G| = 2|H| = 32).

Uses 1

Statement usesBerkovich's problem on automorphism groups of p-groups (Kourovka 15.27)

1 fully proved1 ready to prove

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.